Boundary higher integrability for very weak solutions of quasilinear parabolic equations

Abstract

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the formut−divA(x,t,∇u)=0onΩ×R, where the non-linear structure divA(x,t,∇u) is modelled after the p-Laplace operator. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness ofA(x,t,∇u). These results are also applicable for systems as well as higher order parabolic equations.